In elementary number theory, the lifting the exponent (LTE) lemma provides several formulas for computing the p-adic valuation of special forms of integers. The lemma is named as such because it describes the steps necessary to "lift" the exponent of in such expressions. It is related to Hensel's lemma.
Background
The exact origins of the LTE lemma are unclear; it has only come into focus within the last few decades.[1][2] Despite chiefly featuring in mathematical olympiads, it is sometimes applied to research topics, such as elliptic curves.[3]
Statements
For any integers and positive integers and , where is a prime such that and , the following identities hold:
- When is odd:
- If , .
- If is odd and , .
- When :
- If , .
- If , .
- For all :
- If , .
- If with odd, .
Outline of proof
Base case
The base case when is proven first. Because ,
The fact that completes the proof. The condition for odd is similar.
General case (odd p)
Via the binomial expansion, the substitution can be used in (1) to show that because (1) is a multiple of but not .[1] Likewise, .
Then, if is written as where , the base case gives .
By induction on ,
A similar argument can be applied for .
General case (p = 2)
The proof for the odd case cannot be directly applied when because the binomial coefficient is only an integral multiple of when is odd.
However, it can be shown that when by writing where is odd and noting that
because each factor in the difference of squares step in the form is congruent to .
The stronger statement when is proven analogously.[1]
In competitions
Example problem
The LTE lemma can be used to solve 2020 AIME I #12:
Let be the least positive integer for which is divisible by Find the number of positive integer divisors of .[4]
Solution: Note that . Using the LTE lemma, since and but , . Thus, . Similarly, but , so and .
Since , the factors of 5 are addressed by noticing that since the residues of modulo 5 follow the cycle and those of follow the cycle , the resiudes of modulo 5 cycle through the sequence . Thus, iff for some positive integer . The LTE lemma can now be applied again: . Since , . Hence .
Combining these three results, it is found that , which has positive divisors.
References
Lifting the Exponent Lemma